Discrete Isoperimetric Inequalities via Curvature

2026-09-21

AI Summary

本文通过基于曲率的半群框架解决了布尔超立方体上的等周不等式问题，适用于弱依赖坐标度量，特别是零场Ising模型。

Abstract

Isoperimetric inequalities on the Boolean hypercube play a fundamental role in the analysis of Boolean functions. These inequalities have been established primarily for the uniform measure and for biased product measures, often through Fourier-analytic or inductive arguments. We develop a curvature-based semigroup framework to establish isoperimetric inequalities for measures with weakly dependent coordinates. In particular, we show that a Dobrushin-type condition, together with marginal boundedness of the distribution, implies the local Bobkov inequality, Talagrand's $L^1$--$L^2$ and variance--surface-area inequalities, the Kahn--Kalai--Linial inequality, and the Eldan--Gross inequality. Our results apply to zero-field Ising models with interaction matrix $J$ throughout the Dobrushin uniqueness regime $\|J\|_1<1$. The framework builds on discrete Bakry--Émery theory and gradient estimates and can also be extended to measures on Hamming slices or hypergrids.

Problem

• Isoperimetric Inequalities
• Boolean Hypercube
• Weakly Dependent Coordinates
• Dobrushin-type Condition
• Bakry-Émery Theory

Innovation

• curvature-based semigroup framework
• isoperimetric inequalities
• Dobrushin-type condition
• weakly dependent coordinates